Corollaries
A special case of the theorem is computing integral cohomology. For a finite CW complex X, is finitely generated, and so we have the following decomposition.
Where are the betti numbers of X. One may check that
- , and
This gives the following statement for integral cohomology:
For X a compact, closed, and connected n-manifold, this corollary coupled with Poincaré duality gives that .
Read more about this topic: Universal Coefficient Theorem